# The observation algebra of spatial pomsets

## Abstract

For sequential programming, the theory of functions provides a uniform metalanguage to describe behaviours by abstracting from the actual implementation of programs. For concurrent and distributed systems, instead, there is no well accepted metalanguage to describe the possible observations of the behaviour of programs. The proper treatment of observations is thus an important and complex issue of concurrency theory. In this paper we show that observations can be described in a uniform way by introducing certain algebras called *observation algebras*. They lift to an algebraic level the standard treatment of actions in the operational semantics of process algebras. Observations are described as terms of an algebra. As a consequence, we separate the *control* level (the operational semantics) from the *data* level (the observations). The chosen notion of observability can be obtained by suitably axiomatizing the operations of the observation algebra. We show how observation algebras can be naturally derived from process algebras. As a case study we consider Milner's CCS. We introduce an observation algebra for CCS and we show that the standard interleaving semantics can be obtained by axiomatizing the operations to yield actions. Furthermore, we give a complete axiomatization of an observation algebra whose elements are certain labelled partial orderings of events (pomsets) called *Spatial Pomsets*.

## Keywords

Normal Form Operational Semantic Sequential Composition Process Algebra Dynamic Logic## Preview

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## References

- [AB84]Austry, D. Boudol, G.
*Algebra de Processus et Synchronization*, Theoretical Computer Science**30**(1), pp 91–131, 1984.Google Scholar - [BC88]Boudol, G., Castellani, I.,
*Concurrency and Atomicity*, Theoretical Computer Science**59**(1,2), pp. 25–84, 1988.Google Scholar - [BC89]Boudol, G., Castellani, I.,
*Permutations of Transitions: An Event Structure Semantics for CCS*, In Proc. REX School, Workshop on Linear Time Branching Time and Partial Orders in Logics and Models for Concurrency, LNCS**354**, pp 411–437 1989.Google Scholar - [BC90]Boudol, G., Castellani, I.,
*Three Equivalent Semantics Semantics for CCS*, In Semantics of Systems of Concurrent Processes, LNCS**469**, pp 96–141, 1990.Google Scholar - [BCKH91]Boudol, G., Castellani, I., Kiehn, A., Hennessy, M.,
*Observing Localities*, To appear Proc. MFCS 1991, LNCS, 1991.Google Scholar - [BHR84]Brookes, S.D., Hoare, C.A.R., Roscoe, A.D.,
*A Theory of Communicating Sequential Processes*, Journal of ACM**31**(3), pp 560–599, 1984.Google Scholar - [BK84]Bergstra, J., Klop, W.,
*Process Algebra for Synchronous Communication*, Information and Control**60**, pp 109–137, 1984.Google Scholar - [CES86]Clarke, E., Emerson, A., Sistla, P.
*Automatic Verifications of Finite State Concurrent Systems using Temporal Logic Specifications*, ACM, TOPLAS,**8**, pp 244–263, 1986.Google Scholar - [CFM90]Corradini, A., Ferrari, G., Montanari, U.
*Transition Systems with Algebraic Structure as Models of Computations*, In Semantics of Systems of Concurrent Processes, LNCS**469**, pp 185–222, 1990.Google Scholar - [DM87]Degano, P., Montanari, U.,
*Concurrent Histories: A Basis for Observing Distributed Systems*, Journal of Computer and System Sciences**34**, pp 422–461, 1987.Google Scholar - [DeS85]De Simone, R.,
*Higher Level Synchronising Devices in MEIJE-SCCS*, Theoretical Computer Science,**37**(3), pp 245–267, 1985.Google Scholar - [FGM91]Ferrari, G., Gorrieri, R., Montanari, U.,
*An Extended Expansion Theorem*, In Proc. TAPSOFT 91, LNCS**494**, pp 29–48, 1991.Google Scholar - [FM90]Ferrari, G., Montanari, U.,
*Towards the Unification of Models for Concurrency*, Proc. CAAP'90, LNCS**431**, pp 162–176, 1990.Google Scholar - [Ha84]Harel, D.
*Dynamic Logic*, In Handbook of Philosophical Logic, Vol II, (Gabbay-Guenthner Eds.), pp 497–604, 1984.Google Scholar - [Ho85]Hoare, C.A.R.,
*Communicating Sequential Processes*, Prentice Hall, 1985.Google Scholar - [He88]Hennessy, M.
*An Algebraic Theory of Processes*, MIT Press, 1988.Google Scholar - [HM85]Hennessy, M., Milner, R.,
*Algebraic Laws for Nondeterminism and Concurrency*, Journal of ACM**32**(1), pp 137–141, 1985.Google Scholar - [La83]Lamport, L.
*What Good is Temporal Logic*, IFIP-83, pp 657–668, 1983.Google Scholar - [Ma77]Mazurkiewicz, A.
*Concurrent Program Schemes and their Interpretations*, Technical Report DAIMI PB-78, Aarhus University, 1977.Google Scholar - [Ma89]Mazurkiewicz, A.
*Basic Notions of Trace Theory*,In Proc. REX School, Workshop on Linear Time Branching Time and Partial Orders in Logics and Models for Concurrency, LNCS**354**, pp 285–363, 1989.Google Scholar - [Mil80]
- [Mil81]Milner, R.
*A Modal Characterization of the Observable Machine Behaviour*, In Proc. CAAP 81, LNCS**112**, pp 25–34, 1981.Google Scholar - [Mil89]Milner, R.,
*Communication and Concurrency*, Prentice Hall, 1989.Google Scholar - [MP89]Manna, Z., Pnueli, A.
*The Anchored Version of the Temporal Framework*, In Proc. REX School, Workshop on Linear Time Branching Time and Partial Orders in Logics and Models for Concurrency, LNCS**354**, pp 201–284 1989.Google Scholar - [Par81]Park, D.,
*Concurrency and Automata on Infinite Sequences*, in Proc. GI, LNCS**104**, pp 167–183, 1981.Google Scholar - [NPW81]Nielsen, M., Plotkin, G., Winskel, G.
*Petri Nets, Event Structures and Domains*(Part I). Theoretical Computer Science,**13**(1), pp 85–108, 1981.Google Scholar - [Plo81]Plotkin, G.
*A Structured Approach to Operational Semantics*, DAIMI FN-19, Computer Science Dept. University of Aarhus, 1981.Google Scholar - [PP90]Plotkin, G., Pratt, W.,
*Teams Can See Pomsets*, Preliminary Report, August 1990.Google Scholar - [Pr86]Pratt, V.,
*Modelling Concurrency with Partial Orders*, Int. Journal of Parallel Programming,**15**(1), pp 381–400, 1986.Google Scholar - [Re85]Reisig, W.
*Petri Nets: An Introduction*, EATCS Monograph, Springer, 1985.Google Scholar - [Sti89]Stirling, C.
*Modal and Temporal Logics*, To appear in Handbook of Logic in Computer Science, Vol I, (Gabbay-Maibaum Eds.) Oxford University Press.Google Scholar - [Win82]Winskel, G.
*Event Structure Semantics for CCS and Related Languages*, Proc. ICALP 82, LNCS**140**, pp 561–576, 1982.Google Scholar